Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities
Entity
UAM. Departamento de MatemáticasPublisher
ElsevierDate
2023-03-15Citation
10.1016/j.jfa.2022.109831
Journal of Functional Analysis 284.6 (2023): 109831
ISSN
0022-1236 (print); 1096-0783 (online)DOI
10.1016/j.jfa.2022.109831Project
info:eu-repo/grantAgreement/EC/H2020/839749/EU//techFRONT; Gobierno de España. CEX2019-000904-S; info:eu-repo/grantAgreement/EC/H2020/777822/EU//GHAIAEditor's Version
https://doi.org/10.1016/j.jfa.2022.109831Subjects
Boundedness Estimates; Gagliardo-Nirenberg-Sobolev Inequalities; Green Functions; Nonlinear Degenerate Parabolic Equations; MatemáticasRights
© 2022 The Author(s). Published by Elsevier Inc.
Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
Abstract
We establish boundedness estimates for solutions of generalized porous medium equations of the form ∂tu+(−L)[um]=0in RN×(0,T), where m≥1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise L1–L∞-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of −L and I−L. In the linear case m=1, it is well-known that the L1–L∞-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques. We establish a similar scenario in the nonlinear setting m>1. First, we can show that operators for which ultracontractivity holds, also provide L1–L∞-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like −L=I−J⁎. They do not regularize when m=1, but we show that surprisingly enough they do so when m>1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator. Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration
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Google Scholar:Bonforte, Matteo
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Endal, Jørgen
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